Dont use the whites. Leave the egg out or cook the yolk, either partially or completely. I sometimes use fully cooked yolks (when cooking for elderly folks -- to avoid any salmonella issues) and that really cuts doen on the eggy taste.

The egg yolk is there for taste, texture (body) and emusification. Using an egg white would be sort of pointless.

You can also use Egg Beaters or something similar - cuts out ALL concerns.

I use the whole egg or fewer yolks than whites - mine has always turned out just fine and I have never noticed an eggy taste. The egg is just a means to get the other ingredients to mix in easily with the lettuce.

The original Caesar salad was made with whole eggs and I myself prefer it this way and if it's just a small group of people that I know I just use raw without the coddling - but I would also much prefer egg beaters to mayo. Mayo has a very definitive flavor and totally changes the taste while the egg beaters at least runs along the same lines as the whole egg. With mayo you just have a creamy mayo-based dressing. I have also, for dietary purposes of some guests, used 1 yolk and the rest egg whites - what you get is the flavor of the other ingredients and the egg white without the yolk can carry those flavor quite nicely. The original recipe also didn't have any anchovies - but I think they add a wonderful layer of flavor.

Re. eggs--From Snopes.com: "Caesar salads have no connection whatsoever to Julius Caesar, or indeed to any of the Caesars who ruled Rome and her far-flung empire. It instead honors Caesar Cardini, a famed restaurateur who invented the dish in Tijuana, Mexico, in 1924. His original recipe called for romaine, garlic, croutons, Parmesan cheese, boiled eggs, olive oil, and Worcestershire sauce. He was said to be staunchly against the inclusion of anchovies in the mixture, contending that Worcestershire sauce is adequate to provide the faint fishy flavor."